Springer Geochemistry - Springer Link

Springer Geochemistry

More information about this series at http://www.springer.com/series/13486

Vojtěch Janoušek Jean-François Moyen Hervé Martin Vojtěch Erban Colin Farrow •

•

Geochemical Modelling of Igneous Processes – Principles And Recipes in R Language Bringing the Power of R to a Geochemical Community

123

Vojtěch Janoušek Czech Geological Survey Prague Czech Republic

Vojtěch Erban Czech Geological Survey Prague Czech Republic

Jean-François Moyen Université Jean-Monnet Saint-Etienne France

Colin Farrow Glasgow Scotland

Hervé Martin Université Blaise-Pascal Clermont-Ferrand France

Springer Geochemistry ISBN 978-3-662-46791-6 DOI 10.1007/978-3-662-46792-3

ISBN 978-3-662-46792-3

(eBook)

Library of Congress Control Number: 2015937379 Springer Heidelberg New York Dordrecht London © Springer-Verlag Berlin Heidelberg 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media (www.springer.com)

Preface

Because I have known the torment of thirst, I would build a well where others may drink.

––– Ernest Thompson Seton

The goal of generations of igneous geochemists is to use mineralogical and chemical laws in an attempt to explain the geological processes they are investigating. This scientific approach is both simple and rigorous. Initially it consists of highlighting magmatic differentiation trends and determining the possible underlying petrogenetic mechanism(s). Then, the major elements are used to establish the nature and the modal composition of the fractionating mineral assemblage responsible for the differentiation trends; its temporal evolution is also addressed. Finally all these data are fed into models calculating the behaviour of trace elements (and possibly isotopes), in order to account for the chemistry of the investigated igneous rocks and evolution of the parental magma. Such methodology is very powerful; not only because it is consistent with field geological data but also it is based on several independent methods. Indeed, major elements, trace elements and isotopes are governed by different principles. Thus any model predicting the coherent behaviour of these three independent parts of the dataset would possess a high internal consistency, making the modelled scenario robust. Everyone, needing to interpret whole-rock geochemical data from igneous rocks, faces the same problem. Regardless of whether he/she has to calculate some simple indexes, more complex norms, plot a diagram for a paper or model effects of some petrogenetic process , he will end up using a computer. He would be certainly delighted to find that several programs exist designed specifically for this purpose. At first glance, most look useful with a plethora of built-in functions, but after a second look, he realizes that they are essentially black boxes, in which he soon loses track of exactly what is happening with his precious data. Worse still, there could be something missing or not quite appropriate to the required task. The code is difficult or impossible to alter (many geochemical programs are commercial). And even when the required diagram is plotted correctly, it may need to be altered extensively before reaching publication quality. Indeed, graphical and numerical methods remain the alpha and omega of modern igneous geochemistry. The problem is how to implement the necessary diagrams or formulae so that the code can be understood and used by an ordinary v

vi

Preface

geochemist. We strongly believe that this knowledge can be mediated in the form of simple numerical recipes in a high-level programming language that includes built-in mathematical and statistical functionality, matrix manipulation tools and be capable of generating publication-quality graphics. There are currently available several potentially suitable environments, but only one of them—the R language (www.r-project.org)—has the advantage of being freely available for all the main platforms (MS Windows, Mac OS and various dialects of Linux). Moreover, there already exists an R package GCDkit (www.gcdkit.org)1, containing most of the required geochemical calculation procedures and graphics. Furthermore , the underlying code can be easily viewed, modified or extended. In the realm of geochemical modelling, there does not exist any prescribed scenario. In fact, the modelling strategy not only depends on the geological problem, but also on the nature of the available data: hence the approach must be adapted and optimized to each individual case. The purpose of this book is to show, using many concrete examples, how a researcher can proceed in developing a realistic model tailored to his questions. It is in this investigative adventure that the authors of this book invite you. Let’s embark on a scientific journey in the intimacy of petrogenetic modelling! Book structure—how to read? This textbook gives a detailed overview of modelling approaches to petrogenesis of igneous rocks using whole-rock geochemical data. The theoretical chapters are followed by their implementation using R/GCDkit, and by numerous exercises, mostly based on real-life problems. The text is divided into six parts, and three appendices. Part I gives a short but comprehensive introduction to R (with, or without GCDkit), the implementation of simple geochemical computations, calculation of norms, statistical evaluation of complex data sets, and plotting the most common diagrams. In all cases, the geological and geochemical backgrounds are briefly discussed. Moreover, a refresher on radiogenic isotope data interpretation is presented. For newbies, the fundamentals and syntax of the R language are explained in Appendix A, and an introduction to the GCDkit system is given in Appendix B. The core of the book (parts II–IV) is dedicated to modelling of the main processes in igneous petrogenesis using various types of geochemical data. These include major elements (treated by the concept of mass balance), trace elements (modelling based on solid/liquid partitioning or saturation concepts) and radiogenic isotopic data (either constraining open-system processes such as mixing and assimilation or giving direct information on the source). The principles of forward and reverse numerical techniques are presented and explained, as is the underlying mathematical apparatus; the R code necessary for their implementation is also given. The specific problem of solving sets of linear algebraic equations is outlined in Appendix C.

1

Natively for Windows, but can be run on other platforms with a suitable emulator environment.

Preface

vii

Part V provides a practical guide on how to formulate and run a sensible petrogenetic model simulating natural systems. It stresses the fundamental significance of additional information coming, e.g., from field relations, petrology or physics. Above all, the importance of critical thinking is underlined. The text is supplemented by numerous solved exercises. It is crowned by two worked real-world problems (Part VI) that illustrate the complex approach to petrogenetic modelling based on the techniques described in this book. On the other hand, intentionally omitted are most of the more sophisticated statistical methods as these have been dealt with by other, more competent authors. This is also the case for detailed mathematical derivations of laws governing geochemical variations in complex petrogenetic scenarios. The book is intended for senior undergraduate and postgraduate courses, as well as all potential users of R/GCDkit interested in the implementation of graphical, statistical and numerical methods. The prerequisite is a sound knowledge of secondary school maths as well as of basic principles of solid-rock geochemistry. Electronic supplementary material Errata, code to the exercises and data sets from this book are available on: http://book.gcdkit.org. Moreover, this web site also contains the scripts used to produce many of the figures. However, in the latter case the code is not always simple and easy to comprehend by a beginner. It is supplied purely for the sake of curiosity, and in order to stimulate the interested reader. They are unlikely to work without at least some adaptations. If reading an electronic version of this book, the exercises, dataset icons and relevant figures are clickable. Most of the exercises in this book are designed to run in an interactive mode. To adopt them for batch use, the contents of any variable should be displayed using the functions print or cat (see Appendix A, Sect. 3.1). The code supplied, obviously, will run only if the current R directory is that in which the data file(s) reside. The best is probably to save all the needed files in a directory of your choice and, before starting, set the working directory either from the GUI (File|Change dir…), or with a command such as: GCDkit->Rbook.dirsetwd(Rbook.dir)

This text is based on version 2.13 of R for Windows, 4.0 of GCDkit. It concentrates on MS Windows implementation of the R language. Plain R will run on other systems, including Linux and Mac OS, but the current GCDkit will require a suitable emulation environment, e.g. Wine on Linux. The code, relying on GCDkit functions, will be displayed with the namesake prompt, GCDkit->.

2

Backslash is an escape character in R, so it would need to be preceded by another one, i.e.:

"C:\\user\\my_name\\Documents\\Rbook".

viii

Preface

Acknowledgements We would like to thank several people without whom this book would have hardly originated. First of all Robert Gentleman and Ross Ihaka, as well as members of the R Development Core Group, for their ideas and sterling efforts in developing the R language. In particular, Friedrich Leisch (Technische Universität, Vienna) provided valuable consultations on the development of R packages. This book builds on hand-outs used for workshops given in Prague and Helsinki (2011), Stellenbosch, South Africa (2012) and Hyderabad, India (2013) by VJ and JFM. The authors are indebted to many people without whom these events would not materialize, especially Tapani Ramö (Helsinki), Gary Stevens (Stellenbosch) and Santosh Kumar (Nainital, India). VJ is grateful to Graeme Rogers, once supervisor of his PhD. thesis, who has taught him to ask the basic petrogenetic questions, why and how much, Francis Albarède for his excellent book and a motivating course of geochemical modelling and Milan Novák (Brno, Czech Republic) with František V. Holub (Prague), who requested, in 2000, a practical course on interpretation of geochemical data in R that is running ever since. GCDkit originated during a post-doctoral stay of VJ at University of Salzburg (FWF Project 15133–GEO to F. Finger). The work was, over the years, supported by several projects from the Czech Grant Agency (GACR), Czech Geological Survey (3314, 336200) and Ministry of Education of the Czech Republic (LK11202 to K. Schulmann). The scientific exchanges were facilitated by the French–Czech programme Mobility 7AMB13FR026. Régis Doucelance (Clermont-Ferrand, France) brought to our attention a great reverse mixing hyperbola example from Martinique, and other papers on mantle geochemistry that escaped our attention. Nice photos were contributed by Gerhard Wörner (Göttingen, Germany), Christian Nicollet (Clermont-Ferrand), Ewa Słaby (Warsaw, Poland) and S. Hidalgo (Quito, Ecuador). Didier Laporte (ClermontFerrand) supplied an original version of his figure on wetting angles (Fig. 24.1). Leon Bagas (Crawley, Australia) provided dataset for the anomaly plot. Oscar Laurent (Liège, Belgium) and Arnaud Villaros (Orléans, France)—in addition to always being pleasant company—are responsible for many good ideas and suggestions, which eventually helped to develop the existing code. We are also grateful to the many students and GCDkit users, whose requests, in the course of time, led to more software development, clever ideas and debugging. We are thankful to people from Springer, Annett Buettner, Ulrike Stricker and Chris Bendall, for their hard work and patience with which they have guided us through the dire straits of preparing and publishing this monograph. The Czech brewing industry was a source of inspiration during late-night coding sessions. In Prague, St. Etienne, Clermont-Ferrand and Glasgow, Christmas 2014 Vojtěch Janoušek ([email protected]) Jean-Francoiú Moyen ([email protected]) Hervé Martin ([email protected]) Vojtěch Erban ([email protected]) Colin Farrow ([email protected])

Typographic Conventions

Most important are the warning boxes indicating potential pitfalls: ‘Warning’

Pointers and additional information opening new prospects (not dealt with in this text) are labelled as: ‘Next step’

The GCDkit implementation of the given problem is outlined in: ‘GCDkit box’

The text is supplemented by a large number of solved exercises, graphically introduced like this: Exercise 1.2: Fractional crystallization

An associated data file is marked by exercise by

, and the beginning of a solution to the

.

In the main text, R code and its output are set in a non-proportional font, the latter additionally in italics. Plain R code has a simple prompt, the GCDkit-specific one is marked as such: > summary(x[,"Sr"]) Min. 1st Qu. Median 278.0 392.5 430.0

Mean 3rd Qu. 443.0 537.5

Max. 599.0

NA's 2.0

GCDkit-> loadData("sazava.data") GCDkit-> results plot(WR[,"SiO2"), + pch="*",col="khaki")

In Part VI and Appendix A, commented chunks of R code, often just outlined and not complete, are displayed as “code boxes”: We create Harker plots (Fig. 25.3) using Plot|Multiple plots…, i.e. the command: GCDkit-> multiple("SiO2","Al2O3,Fe2O3,MgO,CaO,Na2O,K2O")

Names for R objects and file names, occurring in the text, are also set in a nonproportional font: … factor silica from the previous exercise … Comments in the code start with the hash mark (“#”): # Fig. 4.2.1

Names of mathematical variables and Menu items are type set in italics, e.g. Misc|Stop current computation, indicating an item ‘Stop current computation’ of the ‘Misc’ menu. Equations in the text are numbered sequentially, starting with the chapter number:

CL

C0 D F (1 D)

(5.1)

The same applies to figures, tables, or exercises. Otherwise, the first number of figures or tables is that of the relevant Appendix; the second refers to a sequence there in (Fig. A1.2; Table A3.5). Matrices are named with a bold italics letter topped by a double bar. When presented in full, they are enclosed in brackets:

C

§ CPlSiO2 ¨ TiO2 ¨ CPl ¨ # ¨¨ P2O5 © CPl

SiO2 COpx TiO2 COpx # P2O5 COpx

} CnSiO2 · ¸ } CnTiO2 ¸ % # ¸ ¸ } CnP2O5 ¸¹

Vectors are written in a similar way, but their symbol has a single arrow:

JJG m

§ mPl · ¨ ¸ ¨ mOpx ¸ ¨ # ¸ ¨ ¸ © mn ¹

xi

Typographic Conventions

Equation systems are linked together by a single curly brace to their left:

SiO2 ° CS ° ° TiO2 ° CS ® ° ° ° P2O5 °CS ¯

n

¦( m C i

SiO2 i

)

i 1 n

¦( m C i

TiO2 i

)

i 1

} n

¦( m C i

i 1

P2O5 i

)

Variables and Symbols

This book uses, as much as possible, a consistent set of symbols to represent variables of geochemical interest both in the text and in the accompanying R code. The following is a summary of the main symbols used. Often, a series of similar variables exists for a range of elements, minerals or portions of a system: this is indicated by subscripts and superscripts. General Symbol

Meaning and definition

R symbol

W

Mass (weight) of a whole system, or a portion thereof (indicated by subscript, see separate table below)

Not used3

wD

The mass of an element Į in a portion of the system (see subscript)

Not used

F

Melt fraction: F WL W0

ff

FC

Degree of crystallization: FC 1 F WS W0

fc

C, CĮ

3

Concentration of element Į in a system or portion thereof (subscript): CD

wD W

c0, cs, cl…

ciD

Concentration of element Į in mineral i

mins

T

Temperature, normally in K but some equations use °C

tt

Some equations discussed in the text are not implemented in the R code. xiii

xiv


Mineral proportions in various portions of a system (always summing up to 1) Symbol


R symbol

mi

… in the whole solid (cumulate or restite)

m

qi

… in the peritectic assemblage

Not used

pi

… in the reactants (non-modal melting)

Not used

m0,i

… in the original solid (non-modal melting)

Not used

Symbol


R symbol

f1, f2…fm

Fraction of end-members 1, 2, …m involved in mixing

a, b, u, v

Various elements for which mixing is modelled (Sect. 11.3)

Not used

A, B, C, D

Parameters of a mixing hyperbola (Sect. 11.3)

AA, BB, CC, DD

Mixing

f1, f2.. or f[1], f[2]…

Element partitioning Symbol

Meaning and definition Partition coefficient of element Į between mineral (min) i and liquid:

K DD min / L

K DD min / L

D cmin CLD

R symbol kd, or commonly kd[j,i] or

kd[elt,min] assuming elt and min have been deOften just KD in text, when its meaning (element/mineral) fined before… is clear from the context Bulk distribution (solid/liquid) of element Į

DD

DD

CSD CLD

n

¦m K D i

i /L D

i

dd dd[elt] dd[j]

Often just D in text when the meaning is unambiguous “Bulk distribution coefficient” for the initial melting assemblage (non-modal melting): n

D0D

D0D

¦m

0i

K DD i /L

Not used

i 1

(or just D0 when unambiguous) “Bulk distribution coefficient” for the reactants (nonmodal melting): n

DPD

DPD

¦pK D i

i/ L

D

i 1

(or just DP when unambiguous)

Not used

xv


AFC Symbol


R symbol

x

WA

Rate of assimilation

Not used

x

WC r

Rate of fractional crystallization Rate of assimilation to fractionation: r

x

WA x

Not used r

WC

rC

Critical value of r for AFC, above which assimilation is dominant

z

Convenience parameter: z

S

Slope of a mixing array in a diagram 1/c vs. I employed in reverse AFC modelling

Not used

C0 , I0

Element concentration and isotopic ratio in pristine melt

c0, i0

CA , IA

Element concentration and isotopic ratio in assimilant

ca, ia

CL , IL

Element concentration and isotopic ratio in liquid

Not used

r D 1 r 1

rc z

Radiogenic isotopes Symbol


R symbol

Ab

Isotopic abundance, e.g. Ab87 Rb

Not used

AW

Atomic weight, e.g., AWRb

Not used

I

Ratio of daughter and stable, unradiogenic isotopes of the I same element (e.g., 87Sr/86Sr, 143Nd/144Nd), measured

IX, I Y

Two such isotopic ratios for two distinct isotopic systems, e.g., Sr and Nd

Not used

R

Ratio of parental isotope/stable, unradiogenic isotope of the daughter element (e.g., 87Rb/86Sr, 147Sm/144Nd)

R

Ȝ

Decay constant

lambda

Ii I1, I2, IM

Initial isotopic ratio (subscripts 1, 2, M indicate isotopic ratios in mixing)

i1, i2, im

X, Y

Isotopic systems if two involved in plotting

e.g., cx1, iy1

b

Slope of an isochron

D

Parameter controlling the shape of a mixing hyperbola

Not used alpha

q

Curvature of a mixing hyperbola

q

x0 , y0

Asymptotes of a mixing hyperbola

x0, y0

xvi


Additional subscripts/superscripts for isotopic ratios SA

Sample

DM

Depleted Mantle

CC

Average Crust

CHUR

Chondritic Uniform Reservoir

0

Present-day

i

Initial

t

At the time t

Parts of a system (subscript) Subscript

Meaning

Process

0

Source

[zero] solid source in the case of melting, primitive magma for crystallization

L

Liquid

L.inst

Instantaneous liquid

L.bulk

Bulk (aggregated) liquid fractional melting

S

Solid

restite for melting, cumulate for crystallization

S.inst

Instantaneous solid

fractional crystallization

S.bulk

Bulk (aggregated) solid

fractional crystallization

M

Mixture

mixing

P

Reactants

non-modal melting

C

Crystallized phases

AFC

A

Assimilant

AFC

Q

Peritectic assemblage

Melting or crystallization (with slightly different meanings)

melt for melting or differentiated liquid for crystallization fractional melting

Additional sub- and superscripts Object

Symbol

Numbering

Minerals

A, B…

i = 1 to n

Chemical elements (components)

Į, ȕ …

j = 1 to p

End-members in a mixing

1, 2…

k = 1 to m

Step of a stepwise process

t

Not used

It is not always possible to use the same conventions for printed text and R variables. For instance, mixed case variables (both upper and lower case) are dangerous in R, because it is case sensitive. Therefore, we use lower case in (most) R variable names. Furthermore, many single-letter symbols are reserved words: for instance c, D and t refer to common R functions (see Appendix A), whereas T and F are shorthand notations for logical TRUE and FALSE. For this reason, the commonly used F (melt fraction) and D (bulk distribution coefficient) are represented by ff and dd in our R code. Note also that the suffix used to indicate the source/parental magma is the number 0, not the letter O!

List of Abbreviations

AFC ASCII BSE CBPC CHUR CL DM ESC FAQ fO2 Ga GUI HFSE ICP-MS INAA ICP-OES ka LA ICP-MS LILE Ma MORB MME NB ppm REE SIMS TIMS TTG XRF

Assimilation and Fractional Crystallization American Standard Code for Information Interchange Back-Scattered Electrons, Mad Cows Disease Central Bohemian Plutonic Complex Chondritic Uniform Reservoir Cathodoluminescence Depleted Mantle Essential Structural Component Frequently Asked Questions oxygen fugacity 109 years Graphical User Interface High-Field Strength Element(s) Inductively Coupled Plasma Mass Spectrometry Instrumental Neutron Activation Analysis Inductively Coupled Plasma Optical Emission Spectrometry 103 years Laser-Ablation Inductively Coupled Plasma Mass Spectrometry Large Ion Lithophile Element(s) 106 years Mid-Ocean Ridge Basalt Mafic Microgranular Enclave Nota Bene Parts Per Million Rare Earth Elements; LREE, MREE, HREE: light, medium, heavy REE Secondary Ion Mass Spectrometry Thermal Ionization Mass Spectrometry Tonalite–Trondhjemite–Granodiorite association X-Ray Fluorescence Spectrometry

xvii

Abbreviations of Mineral Names

Symbols for mineral names mostly follow Kretz (1983): Ab All Amp An Ap Bt Cpx Crd Di En Fo Grt Hbl Ilm Kfs Mnz Ms Mt Ol Opx Phl Pl Qtz Rt Sil Spl Zrn

Albite Allanite Amphibole Anorthite Apatite Biotite Clinopyroxene Cordierite Diopside Enstatite Forsterite Garnet Hornblende Ilmenite Potassium feldspar Monazite Muscovite Magnetite Olivine Orthopyroxene Phlogopite Plagioclase Quartz Rutile Sillimanite Spinel Zircon

Reference Kretz R (1983) Symbols for rock-forming minerals. Amer Miner 68:277-279.

xix

Contents

1 Introduction...................................................................................................... 1 Causes of Whole-Rock Chemical Variation in Igneous Suites .............. 1 1.1 1.2 Conventional Software for Igneous Geochemistry................................. 3 1.2.1 Spreadsheets ............................................................................. 3 1.2.2 Dedicated Programs (PC Compatibles) .................................... 4 1.3 A Revolution? The R Language ............................................................. 4 1.3.1 What is R? ................................................................................ 4 1.3.2 Geochemical Data Toolkit (GCDkit) ....................................... 5 References..........................................................................................................6

Part I R/GCDkit at Work 2 Data Manipulation and Simple Calculations............................................... 11 Loading and Manipulating Data ........................................................... 11 2.1 Linking Whole-Rock Chemistry with Mineral Stoichiometry ............. 13 2.2 Basic Indexes.......................................................................... 14 2.2.1 Cationic Parameters ................................................................ 16 2.2.2 Normative Calculations and Classification 2.2.3 of Igneous Rocks..................................................................... 17 Statistics ............................................................................................... 18 2.3 Classification and Grouping—Using Factors ....................................... 19 2.4 Statistical Examination of Complex Data Sets ....................... 20 2.4.1 Conversion of Numeric Vectors to Factors ............................ 22 2.4.2 Frequency (Contingency) Tables ........................................... 23 2.4.3 References........................................................................................................ 24 3 Classical Plots ................................................................................................27 3.1 Binary Plots .......................................................................................... 27 Plotting Simple Binary Plots .................................................. 27 3.1.1 Constant Sum Effect (Closure) ............................................... 30 3.1.2 Harker Plots and Other Basic Multiple Plots ........................................ 31 3.2 Ternary Plots ........................................................................................ 32 3.3 Classification Plots in GCDkit . ............................................................ 34 3.4 Geotectonic Diagrams .......................................................................... 35 3.5 Spiderplots ............................................................................................ 36 3.6 Multiple Plots by Groups ...................................................................... 40 3.7 References........................................................................................................ 41 xxi

xxii

Contents

4 Specialized Plots ............................................................................................. 45 4.1 Log–Log Binary Plots .......................................................................... 45 4.2 Specialized Spiderplots ........................................................................ 46 Double-Normalized Spiderplots ............................................. 47 4.2.1 4.2.2 Spider Boxplots, Spider Box and Percentile Plots .................. 47 Contour Plots .............. .......................................................................... 48 4.3 Anomaly Plots....................................................................................... 49 4.4 Stripplots and Strip Boxplots.............. .................................................. 50 4.5 References........................................................................................................ 51 5 Radiogenic Isotopes ....................................................................................... 53 5.1 Recalculation of Elemental to Isotopic Ratios ...................................... 53 5.2 Calculation of Initial Ratios or Ages .................................................... 55 5.3 Epsilon, Delta and Gamma Values ....................................................... 57 5.4 Model Ages .......................................................................................... 60 5.4.1 Single-Stage Nd Model Ages ................................................. 60 5.4.2 Two-Stage Nd Model Ages .................................................... 61 5.5 Isochron Ages ....................................................................................... 63 References........................................................................................................ 65

Part II Modelling Major Elements 6 Direct Models ..................................................................................................69 6.1 Mass Balance During Crystallization ................................................... 69 6.1.1 Graphical Solutions ................................................................ 70 6.1.2 Cumulate Composition ........................................................... 71 6.1.3 Analytical Formulation ........................................................... 72 6.1.4 Generating a Magmatic Series Through Crystallization ......... 73 6.2 Partial Melting ...................................................................................... 75 6.3 Peritectic Reactions .............................................................................. 76 6.4 Mixing................................................................................................... 78 6.5 Crystallization and Partial Melting—Are These Just Special Cases of Mixing?................................................................................. 79 Reference......................................................................................................... 80 7 Reverse Models .............................................................................................. 81 7.1 An Under-Determined Problem............................................................ 81 7.2 Least-Square Solution to Crystallization/Melting Problems ................ 82 7.3 Least-Square Solution to Mixing Problems .......................................... 83 References........................................................................................................ 83 8 Forward Modelling in R ............................................................................... 85 References........................................................................................................ 92 9 Reverse Modelling in R. ................................................................................ 93

Contents

xxiii

Part III Modelling Trace Elements 10 Dilute Trace Elements: Partition Coefficients........................................... 101 10.1 Crystal Networks and Substitutions ................................................... 101 10.2 Partition Coefficients .......................................................................... 101 10.3 Controls on the Values of Partition Coefficients .................................102 10.4 Bulk Distribution Coefficients ........................................................... 103 References...................................................................................................... 104 11 Direct (Dilute) Trace-Element Models ....................................................... 105 11.1 Crystallization ................................................................................... 106 11.1.1 Batch Crystallization .......................................................... 106 11.1.2 Fractional Crystallization ................................................... 106 11.1.3 Comparing Different Models..............................................109 11.2 Melting................................................................................................110 11.2.1 Batch Melting ....................................................................... 110 11.2.2 Fractional Melting ................................................................ 110 11.2.3 Comparing Different Models................................................ 111 11.2.4 Alternative Formulations of the Melting Equations ............. 111 11.3 Mixing.................................................................................................114 11.3.1 Ratio of Two Elements During Binary Mixing .................... 114 11.3.2 Mixing Hyperbolae in Ratio–Ratio Plots ............................. 114 11.4 Assimilation and Fractional Crystallization (AFC) ............................ 117 11.5 Composite Models .............................................................................. 119 11.5.1 Crystallization with Incomplete Crystal Separation (Batch Crystallization + Binary Mixing) ......................................... 120 11.5.2 Fluxed Melting (Binary Mixing + Batch Melting) ............... 121 References...................................................................................................... 123 12 Reverse (Dilute) Trace-Element Models ..................................................... 125 12.1 Reverse Fractional Crystallization (Using Rayleigh’s Law) .............. 125 12.2 Reverse Batch Partial Melting ............................................................ 127 12.3 Reverse Mixing .................................................................................. 128 References...................................................................................................... 128 13 Trace Elements as Essential Structural Constituents of Accessory Minerals: the Solubility Concept.................................................................129 13.1 Solubility Formulae for Common Accessory Minerals ...................... 129 13.1.1 Zircon, ZrSiO4 ...................................................................... 129 13.1.2 Monazite, (LREE)PO4 .......................................................... 131 13.1.3 Apatite, Ca5(PO4)3(F,Cl,OH) ................................................ 134 13.2 Evolution Through Saturation ............................................................ 136 References...................................................................................................... 139 14 Forward Modelling in R ............................................................................. 141 References...................................................................................................... 152 15 Reverse Modelling in R................................................................................ 153

xxiv

Contents

Part IV Radiogenic Isotopes 16 Direct Models ................................................................................................ 159 16.1 Binary Mixing .................................................................................... 159 16.1.1 Single Isotopic Ratio ............................................................ 159 16.1.2 Pair of Isotopic Ratios .......................................................... 161 16.2 AFC Formulation for Isotopes ............................................................ 163 References...................................................................................................... 166 17 Reverse Models ............................................................................................ 167 17.1 Binary Mixing .................................................................................... 167 17.2 AFC .....................................................................................................167 References...................................................................................................... 168 18 Forward Modelling in R ............................................................................. 169 18.1 Binary Mixing .................................................................................... 169 18.2 AFC.....................................................................................................173 19 Reverse Modelling in R ............................................................................... 175 Reference. ...................................................................................................... 177

Part V Practical Modelling 20 Choosing an Appropriate Model ................................................................ 181 20.1 Evidence for Crystallization ............................................................... 181 20.1.1 Final Solidification During Emplacement ............................ 181 20.1.2 Fractional Crystallization at Depth ....................................... 182 20.2 Evidence For Melting ......................................................................... 183 20.2.1 Crustal Anatexis ................................................................... 183 20.2.2 Melting of the Mantle ........................................................... 184 20.3 Magma Mixing and Assimilation ....................................................... 185 20.3.1 Mixing and Mingling of Magmas ......................................... 185 20.3.2 Assimilation.......................................................................... 186 References...................................................................................................... 187 21 Semi-Quantitative Geochemical Approach ............................................... 191 21.1 Assessing Trace-Element Compatibility ............................................ 191 21.2 Order of (In)Compatibility ................................................................. 192 21.3 Process Identification ......................................................................... 195 21.3.1 Mixing vs. Crystallization/Melting....................................... 195 21.3.2 Crystallization vs. Melting in a Log–Log Diagram ............. 195 21.3.3 Crystallization vs. Melting Using Incompatible Elements ... 198 21.4 Mixing Test ........................................................................................ 199 21.5 Identifying Fractionating Minerals Using Log–Log Plots .................. 202 References...................................................................................................... 203 22 Constraining a Model . ................................................................................. 205 22.1 Choosing an Appropriate Strategy...................................................... 205

Contents

xxv

22.2

Constraining a Fractionation or Melting Model ................................. 206 22.2.1 The Differentiated Liquid(s)/Melt: C L ................................. 206 22.2.2 The Primitive Liquid/Source: C0 .......................................... 206 22.2.3 Composition of the Solid Phases (Cumulate/Restite)........... 208 22.2.4 Partition Coefficients ............................................................ 211 22.3 Dealing with Accessory Minerals ....................................................... 212 22.3.1 Assessing the Role of Accessory Minerals........................... 212 22.3.2 Modelling with Accessories ................................................. 214 22.4 Constraining the End-Members of a Mixing Model ........................... 221 References...................................................................................................... 222 23 Numerical Tips and Tricks ......................................................................... 225 23.1 The Size of the Geochemical Space ................................................... 225 23.2 Reducing the System .......................................................................... 227 23.3 Colinearity .......................................................................................... 227 23.4 Breaking Minerals into End Members ............................................... 228 23.5 Coupling Majors and Traces .............................................................. 228 23.6 This Space Left Blank for Your Own Tricks ..................................... 229 References...................................................................................................... 229 24 Common Sense in Action. ........................................................................... 231 24.1 Physical Constraints ........................................................................... 231 24.1.1 Thermodynamic Constraints ................................................ 231 24.1.2 Mechanical (Rheological) Constraints ................................. 232 24.2 Scale and Speed of Processes—Approach to Equilibrium ................. 234 24.3 Is Your Model Worth Your Efforts? .................................................. 235 24.3.1 How Well Can We Discriminate Between Models? ............ 236 24.3.2 Dangerous Projections .......................................................... 237 24.3.3 KD vs. C0 —What Should You Improve First? ..................... 238 24.4 Back to the Field!................................................................................ 239 References...................................................................................................... 240

Part VI Worked Examples 25 Differentiation of a Calc-Alkaline Series: Example of the Atacazo-Ninahuilca volcanoes, Ecuador. ................................................... 245 25.1 Geological Setting .............................................................................. 245 25.2 Data Exploration and Implications ..................................................... 246 25.2.1 Isotopic Data ........................................................................ 246 25.2.2 Major Elements .................................................................... 247 25.2.3 Trace Elements ..................................................................... 248 25.3 Geochemical Modelling ..................................................................... 250 25.3.1 First Step: Atacazo ............................................................... 252 25.3.2 Second Step: Ninahuilca ...................................................... 256 25.4 Summary ............................................................................................ 258 References...................................................................................................... 260

xxvi

Contents

26 Progressive Melting of a Metasedimentary Sequence: the Saint-Malo Migmatitic Complex, France ...................................................................... 261 26.1 Geological Setting .............................................................................. 261 26.2 Major and Trace Elements.................................................................. 262 26.3 Geochemical Modelling ..................................................................... 264 26.3.1 Mode Evolution During melting .......................................... 265 26.3.2 Major and Trace Elements.................................................... 268 26.3.3 Zircon ................................................................................... 271 References...................................................................................................... 275

Appendix A R Syntax in a Nutshell 1 Direct Mode .................................................................................................. 277 1.1 Basic Operations ................................................................................. 277 1.1.1 Starting and Terminating the R Session ............................... 277 1.1.2 Seeking Help and Documentation ........................................ 278 1.2 Fundamental Objects of the R Language............................................ 279 1.2.1 Commands ............................................................................ 279 1.2.2 Handling Objects in Memory ............................................... 281 1.2.3 Attributes to Objects ............................................................. 282 1.3 Numeric Vectors................................................................................. 282 1.3.1 Assignment ........................................................................... 282 1.3.2 Vector Arithmetic ................................................................. 282 1.3.3 Names ................................................................................... 283 1.3.4 Generating Regular Sequences ............................................. 283 1.3.5 Functions to Manipulate Numeric Vectors ........................... 284 1.4 Character Vectors ............................................................................... 284 1.5 Logical Vectors .................................................................................. 285 1.5.1 Logical Operators ................................................................. 285 1.5.2 Missing Values (NA, NaN) .................................................. 286 1.6 Arrays, Matrices, Data Frames ........................................................... 286 1.6.1 Matrix/Data Frame Operations ............................................. 287 1.7 Indexing/Subsetting of Vectors, Arrays and Data Frames.................. 288 Vectors ................................................................................. 289 1.7.1 1.7.2 Matrices/Data Frames ........................................................... 289 1.8 Lists ....................................................................................................290 1.9 Coercion of Individual Object Types ................................................. 291 1.10 Factors.................................................................................................292 1.10.1 Basic Usage of Factors ......................................................... 292 1.10.2 Conversion of Numeric Vectors to Factors .......................... 292 1.10.3 Frequency Tables ................................................................. 293 1.10.4 Using Factors to Handle Complex Datasets ......................... 293 1.11 Data Input/Output, Files ..................................................................... 293 1.11.1 Reading Data ........................................................................ 293 1.11.2 Sample Data Sets .................................................................. 295 1.11.3 Saving Data .......................................................................... 295

Contents

xxvii

2 Graphics........................................................................................................ 295 2.1 Obtaining and Annotating Binary Plots .............................................. 295 2.2 Additional High-Level Plotting Functions ......................................... 302 2.3 Creating Custom Layouts and Axes ................................................... 305 2.4 Exporting Graphs from R and Graphical Devices .............................. 308 2.5 Interaction with Plots.......................................................................... 308 3 Programming in R.........................................................................................309 3.1 Input and Output................................................................................. 309 3.2 Conditional Execution ........................................................................ 310 3.3 Loops.................................................................................................. 310 3.4 User-Defined Functions...................................................................... 311 3.4.1 Arguments to Functions........................................................311 3.4.2 Assignments in Functions ....................................................312 3.5 An Alternative to Loops—sapply . ...................................................... 313 References........ .............................................................................................. 313

Appendix B Introduction to GCDkit 1 First Steps with GCDkit.............................................................................. 315 1.1 Installation .......................................................................................... 315 1.2 GCDkit Overview: The User Interface ............................................... 316 1.3 Working with Data ............................................................................. 318 1.3.1 Data Format .......................................................................... 318 1.3.2 Loading Data ........................................................................ 319 1.3.3 Merging Data ....................................................................... 321 1.3.4 Choosing Data ...................................................................... 322 1.3.5 Grouping............................................................................... 322 1.4 Plotting................................................................................................ 323 1.4.1 Plot Settings ......................................................................... 326 1.4.2 Single Plot Editing ............................................................... 327 1.4.3 Plates and Plate Editing ........................................................ 328 1.4.4 Spiderplots............................................................................ 328 1.4.5 Classification and Geotectonic Diagrams ............................ 329 1.5 Calculations ........................................................................................ 329 1.6 Exporting from GCDkit. ..................................................................... 331 2 Under the Bonnet: GCDkit Internals......................................................... 332 2.1 R language and GCDkit. ..................................................................... 332 2.2 Data Variables .................................................................................... 333 2.3 System Variables ................................................................................ 333 2.4 Tailoring GCDkit to Suit your Needs..................................................334 2.5 Plugins.................................................................................................334 References........................................................................................................336

xxviii

Contents

Appendix C Solving Systems of Linear Algebraic Equations in R 1.1 Linear Algebraic Equation Systems (Single Solution) ....................... 337 1.2 Overdetermined Systems (Unconstrained Least-Square Method)...... 338 1.3 Constrained Least-Square Method ..................................................... 339 References...................................................................................................... 340 Index ................................................................................................................... 341